Soap makes water wetter. Kind of. It reduces the surface tension, allowing it to spread out and penetrate better. Making it clean better. So could we produce a surface tension-free water?

You seem to be assuming that a perfectly tensionless water would make for a superior cleaner, but there’s no reason to assume that would be true. Lowering surface tension is not the sole reason that soap is a good cleaner; it’s not even the biggest reason.
I have no idea if water can be made perfectly tensionless, but that’s what dishwasher rinse-aids are for: to make the water molecules stick to things (including each other) less, so they evaporate more quickly & easily. If You Want Drier Dishes, Use Dishwasher Rinse Aid
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Edit to add: I suppose that definitionally no liquid is without surface tension.
A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Like a gas, a liquid is able to flow and take the shape of a container. Unlike a gas, a liquid maintains a fairly constant density and does not disperse to fill every space of a container.

So could we produce a surface tension-free water?
Homie dats a gas. Or supercritical fluid, which actually is indeed used for "washing" (SC CO2 is used to decaffeinate coffee). However, like others said, surface tension /= cleaning ability. Part of what soap does is increase the effective solubility of things that are not normally soluble.
- Huh!

The soap makes water wetter things was from a Bill Nye episode. I mean he really simplified it for a kids science show. There's a bit more going on with soap than simply the surface tension being reduced. You've got emulsifier, surfactants, etc.
I don't think reducing surface tension alone will make a better cleaner.

The only liquid we've discovered with no surface tension is helium. And only at or below specific temperatures depending on the type of helium.
Very unlikely water would ever get to that state.
- Water has hella intermolecular attraction, unlike noble gases. That's why even really heavy noble elements are in the gas state at room temperature and pressure

The chemical properties that create surface tension help water become a great solvent for a variety of chemicals. There isn't a chemical that wouldn't have surface tension that would also be as good a solvent in that class of materials.

Soap works by attaching one side of molecule to the grease and one side to the water to pull dirt out of something, surface tension is not the only thing helping

https://en.m.wikipedia.org/wiki/Surfactant
The surface tension of water is due to the molecular bonds. It's unlikely to change because then it would no longer be water.

Interesting question. It'd be really neat if you could spray your car with something and have it slide off bringing all of the dirt and dust with it. No need to rinse, even. Hopefully it won't ruin your driveway in the process.
Water and soap has one giant advantage at this point, which is volume. In most populated places water is plentiful.
- It would have to distinguish between dirt and dust and clear coat, paint, primer, and metal.

soap allows to dissolve fats in small droplets of soap (micelles). Surface tension arises from affinity to the surface, like oil still doesn't like teflon pans, despite being hydrophobic. So no, you can't find one size fits all liquid. (Aside from supercritical fluids, but they require pressures - and sometimes are used in dry cleaning! so this might be your answer)

oh and you're talking about dry cleaning chemicals. dry cleaning isn't actually clean, it's just tumbled in chemicals that aren't water

Water isn't wet though.
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  Hehe

The best cleaner that I know of for breaking surface tension is Dawn. From what I understand if enough of it gets into a body of water fish can suffocate because it affects how the water flows around their gills and doesn't allow them to get oxygen.